Question: Simplify the following expression and state the condition under which the simplification is valid: $q = \dfrac{t^2 - 2t - 80}{t^2 + 7t - 8}$
First factor the expressions in the numerator and denominator. $ \dfrac{t^2 - 2t - 80}{t^2 + 7t - 8} = \dfrac{(t - 10)(t + 8)}{(t - 1)(t + 8)} $ Notice that the term $(t + 8)$ appears in both the numerator and denominator. Dividing both the numerator and denominator by $(t + 8)$ gives: $q = \dfrac{t - 10}{t - 1}$ Since we divided by $(t + 8)$, $t \neq -8$. $q = \dfrac{t - 10}{t - 1}; \space t \neq -8$